lin-0161
12.7 From commutators to a Čech calculation
A second KET-related study sharpens the meaning of order sensitivity. Pairwise commutators are placed on the edges of a declared finite nerve, producing a one-cochain. Hodge decomposition separates
Only the harmonic component represents the nontrivial first cohomology class of the declared nerve.
On pretrained models, commutator energy is \(86.76\% \) exact, \(10.31\% \) coexact, and \(2.92\% \) harmonic. Task-gated repair reduces held-out closure, harmonic, and tangent non-exact RMS on all three seeds while test cross-entropy changes by \(-0.03\% \).
Commutator energy is a local obstruction proxy. It becomes a Čech cohomology diagnostic only after a cover or nerve, orientation, cocycle test, and quotient by coboundaries have been declared.
The nerve here has vertices for attention, normalization, feedforward, and geometric incidence transport; changing the nerve changes the cohomology question. There is no claim that a transformer has one canonical Čech cover.